Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{-9x^2 - 99x - 162}{-5x^3 - 75x^2 - 270x}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {-9(x^2 + 11x + 18)} {-5x(x^2 + 15x + 54)} $ $ y = \dfrac{9}{5x} \cdot \dfrac{x^2 + 11x + 18}{x^2 + 15x + 54} $ Next factor the numerator and denominator. $ y = \dfrac{9}{5x} \cdot \dfrac{(x + 9)(x + 2)}{(x + 9)(x + 6)}$ Assuming $x \neq -9$ , we can cancel the $x + 9$ $ y = \dfrac{9}{5x} \cdot \dfrac{x + 2}{x + 6}$ Therefore: $ y = \dfrac{ 9(x + 2)}{ 5x(x + 6)}$, $x \neq -9$